Tabel integral

Kalkulus
Teorema dasar Limit fungsi Kontinuitas Teorema nilai purata Teorema Rolle
Diferensial Definisi Turunan (perumuman) Tabel turunan Diferensial infinitesimal fungsi total Konsep Notasi untuk pendiferensialan Turunan kedua Turunan ketiga Perubahan variabel Pendiferensialan implisit Laju yang berkaitan Teorema Taylor Kaidah dan identitas Kaidah penjumlahan dalam pendiferensialan Perkalian Rantai Pangkat Pembagian Rumus Faà di Bruno
Integral Definisi Antiderivatif Integral (takwajar) Integral Riemann Integrasi Lebesgue Integrasi kontur Tabel integral Integrasi secara parsial cakram kulit tabung substitusi (trigonometri) pecahan parsial Urutan Rumus reduksi
Deret geometri (aritmetika-geometrik) harmonik selang-seling pangkat binomial Taylor Uji kekonvergenan uji suku rasio akar integral perbandingan langsung perbandingan limit deret selang-seling kondensasi Cauchy Dirichlet Abel
Vektor Gradien Divergence Keikalan Laplace berarah identitas Teorema Kedivergenan Gradien Green Stokes
Multivariabel Formalisme matriks tensor eksterior geometrik Definisi Turunan parsial Integral lipat Integral garis Permukaan integral integral volume Jacobi Hesse
Khusus fraksional Malliavin stokastik variasi
l b s

Pengintegralan atau integrasi merupakan operasi dasar dalam kalkulus integral. Operasi lawannya, turunan, mempunyai kaidah yang dapat menurunkan fungsi dengan bentuk yang lebih mudah menjadi fungsi dengan bentuk yang lebih rumit. Sayangnya, integral tidak mempunyai kaidah yang dapat menghitung sebaliknya, sehingga sering kali diperlukan tabel yang memuat kumpulan integral.

Berikut adalah daftar yang memuat integral atau antiturunan yang paling umum dijumpai. Pada daftar di bawah ini, C {\displaystyle C} mengartikan konstanta sembarang.

Daftar integral

Daftar integral yang lebih detail dapat dilihat pada halaman-halaman berikut

• Daftar integral dari fungsi rasional
• Daftar integral dari fungsi irrasional
• Daftar integral dari fungsi trigonometri
• Daftar integral dari fungsi trigonometri terbalik
• Daftar integral dari fungsi hiperbolik
• Daftar integral dari fungsi hiperbolik terbalik
• Daftar integral dari fungsi eksponensial
• Daftar integral dari fungsi logaritmik
• Daftar integral dari fungsi Gaussian

Aturan integrasi dari fungsi-fungsi umum

1. ∫ a f ( x ) d x = a ∫ f ( x ) d x ( a  konstan) {\displaystyle \int af(x)\,dx=a\int f(x)\,dx\qquad {\mbox{(}}a{\mbox{ konstan)}}\,\!}
2. ∫ [ f ( x ) + g ( x ) ] d x = ∫ f ( x ) d x + ∫ g ( x ) d x {\displaystyle \int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx}
3. ∫ f ( x ) g ( x ) d x = f ( x ) ∫ g ( x ) d x − ∫ [ f ′ ( x ) ( ∫ g ( x ) d x ) ] d x {\displaystyle \int f(x)g(x)\,dx=f(x)\int g(x)\,dx-\int \left[f'(x)\left(\int g(x)\,dx\right)\right]\,dx}
4. ∫ [ f ( x ) ] n f ′ ( x ) d x = [ f ( x ) ] n + 1 n + 1 + C (untuk  n ≠ − 1 ) {\displaystyle \int [f(x)]^{n}f'(x)\,dx={[f(x)]^{n+1} \over n+1}+C\qquad {\mbox{(untuk }}n\neq -1{\mbox{)}}\,\!}
5. ∫ f ′ ( x ) f ( x ) d x = ln ⁡ | f ( x ) | + C {\displaystyle \int {f'(x) \over f(x)}\,dx=\ln {\left|f(x)\right|}+C}
6. ∫ f ′ ( x ) f ( x ) d x = 1 2 [ f ( x ) ] 2 + C {\displaystyle \int {f'(x)f(x)}\,dx={1 \over 2}[f(x)]^{2}+C}

Integral fungsi sederhana

Konstanta C sering digunakan untuk konstanta sembarang dalam integrasi. Konstanta ini hanya dapat ditentukan jika suatu nilai integral pada beberapa titik sudah diketahui. Jadi, setiap fungsi mempunyai jumlah integral tidak terbatas.

Rumus-rumus berikut hanya menyatakan dalam bentuk lain pernyataan-pernyataan dalam tabel turunan.

Fungsi rasional

∫ d x = x + C {\displaystyle \int \,dx=x+C}

∫ x n d x = x n + 1 n + 1 + C  jika  n ≠ − 1 {\displaystyle \int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ jika }}n\neq -1}

∫ ( a x + b ) n d x = ( a x + b ) n + 1 a ( n + 1 ) + C  jika  n ≠ − 1 {\displaystyle \int (ax+b)^{n}\,dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad {\mbox{ jika }}n\neq -1}

∫ d x x = ln ⁡ | x | + C {\displaystyle \int {dx \over x}=\ln {\left|x\right|}+C}

∫ d x a 2 + x 2 = 1 a arctan ⁡ x a + C {\displaystyle \int {dx \over {a^{2}+x^{2}}}={1 \over a}\arctan {x \over a}+C}

Fungsi irrasional

∫ d x a 2 − x 2 = arcsin ⁡ x a + C {\displaystyle \int {dx \over {\sqrt {a^{2}-x^{2}}}}=\arcsin {x \over a}+C}

∫ − d x a 2 − x 2 = arccos ⁡ x a + C {\displaystyle \int {-dx \over {\sqrt {a^{2}-x^{2}}}}=\arccos {x \over a}+C}

∫ d x a 2 + x 2 = 1 a arctan ⁡ x a + C {\displaystyle \int {dx \over a^{2}+x^{2}}={1 \over a}\arctan {x \over a}+C}

∫ − d x a 2 + x 2 = 1 a arccot ⁡ x a + C {\displaystyle \int {-dx \over a^{2}+x^{2}}={1 \over a}\operatorname {arccot} {x \over a}+C}

∫ d x x x 2 − a 2 = 1 a arcsec ⁡ | x | a + C {\displaystyle \int {dx \over x{\sqrt {x^{2}-a^{2}}}}={1 \over a}\operatorname {arcsec} {|x| \over a}+C}

∫ − d x x x 2 − a 2 = 1 a arccsc ⁡ | x | a + C {\displaystyle \int {-dx \over x{\sqrt {x^{2}-a^{2}}}}={1 \over a}\operatorname {arccsc} {|x| \over a}+C}

Fungsi eksponensial

∫ e x d x = e x + C {\displaystyle \int e^{x}\,dx=e^{x}+C}

∫ a x d x = a x ln ⁡ a + C {\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C}

Fungsi logaritma

∫ ln ⁡ x d x = x ln ⁡ x − x + C {\displaystyle \int \ln {x}\,dx=x\ln {x}-x+C}

∫ b log ⁡ x d x = x ⋅ b log ⁡ x − x ⋅ b log ⁡ e + C {\displaystyle \int \,^{b}\!\log {x}\,dx=x\cdot \,^{b}\!\log x-x\cdot \,^{b}\!\log e+C}

Fungsi trigonometri

Artikel utama: Daftar integral dari fungsi trigonometri

∫ sin ⁡ x d x = − cos ⁡ x + C {\displaystyle \int \sin {x}\,dx=-\cos {x}+C}

∫ cos ⁡ x d x = sin ⁡ x + C {\displaystyle \int \cos {x}\,dx=\sin {x}+C}

∫ tan ⁡ x d x = ln ⁡ | sec ⁡ x | + C {\displaystyle \int \tan {x}\,dx=\ln {\left|\sec {x}\right|}+C}

∫ cot ⁡ x d x = − ln ⁡ | csc ⁡ x | + C {\displaystyle \int \cot {x}\,dx=-\ln {\left|\csc {x}\right|}+C}

∫ sec ⁡ x d x = ln ⁡ | sec ⁡ x + tan ⁡ x | + C {\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C}

∫ csc ⁡ x d x = − ln ⁡ | csc ⁡ x + cot ⁡ x | + C {\displaystyle \int \csc {x}\,dx=-\ln {\left|\csc {x}+\cot {x}\right|}+C}

∫ sec 2 ⁡ x d x = tan ⁡ x + C {\displaystyle \int \sec ^{2}x\,dx=\tan x+C}

∫ csc 2 ⁡ x d x = − cot ⁡ x + C {\displaystyle \int \csc ^{2}x\,dx=-\cot x+C}

∫ sec ⁡ x tan ⁡ x d x = sec ⁡ x + C {\displaystyle \int \sec {x}\,\tan {x}\,dx=\sec {x}+C}

∫ csc ⁡ x cot ⁡ x d x = − csc ⁡ x + C {\displaystyle \int \csc {x}\,\cot {x}\,dx=-\csc {x}+C}

∫ sin 2 ⁡ x d x = 1 2 ( x − sin ⁡ x cos ⁡ x ) + C {\displaystyle \int \sin ^{2}x\,dx={\frac {1}{2}}(x-\sin x\cos x)+C}

∫ cos 2 ⁡ x d x = 1 2 ( x + sin ⁡ x cos ⁡ x ) + C {\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}(x+\sin x\cos x)+C}

∫ sec 3 ⁡ x d x = 1 2 sec ⁡ x tan ⁡ x + 1 2 ln ⁡ | sec ⁡ x + tan ⁡ x | + C {\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C}

∫ sin n ⁡ x d x = − sin n − 1 ⁡ x cos ⁡ x n + n − 1 n ∫ sin n − 2 ⁡ x d x {\displaystyle \int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}

∫ cos n ⁡ x d x = cos n − 1 ⁡ x sin ⁡ x n + n − 1 n ∫ cos n − 2 ⁡ x d x {\displaystyle \int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}

Fungsi trigonometri terbalik

Artikel utama: Daftar integral dari fungsi trigonometri terbalik

∫ arcsin ⁡ ( x ) d x = x a r c s i n ( x ) + 1 − x 2 + C {\displaystyle \int \arcsin(x)\,dx=x\,arcsin(x)+{\sqrt {1-x^{2}}}+C}

∫ arccos ⁡ ( x ) d x = x a r c c o s ( x ) − 1 − x 2 + C {\displaystyle \int \arccos(x)\,dx=x\,arccos(x)-{\sqrt {1-x^{2}}}+C}

∫ arctan ⁡ x d x = x arctan ⁡ x − 1 2 ln ⁡ | 1 + x 2 | + C {\displaystyle \int \arctan {x}\,dx=x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}

∫ arccot ⁡ x d x = x arccot ⁡ x + 1 2 ln ⁡ | 1 + x 2 | + C {\displaystyle \int \operatorname {arccot} {x}\,dx=x\,\operatorname {arccot} {x}+{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}

∫ arcsec ⁡ ( x ) d x = x arcsec ⁡ ( x ) − ln ⁡ ( | x | + x 2 − 1 ) + C = x arcsec ⁡ ( x ) − arcosh ⁡ | x | + C {\displaystyle \int \operatorname {arcsec}(x)\,dx=x\operatorname {arcsec}(x)\,-\,\ln \left(\left|x\right|+{\sqrt {x^{2}-1}}\right)\,+\,C=x\operatorname {arcsec}(x)-\operatorname {arcosh} |x|+C}

∫ arccsc ⁡ ( x ) d x = x arccsc ⁡ ( x ) + ln ⁡ ( | x | + x 2 − 1 ) + C = x arccsc ⁡ ( x ) + arcosh ⁡ | x | + C {\displaystyle \int \operatorname {arccsc}(x)\,dx=x\operatorname {arccsc}(x)\,+\,\ln \left(\left|x\right|+{\sqrt {x^{2}-1}}\right)\,+\,C=x\operatorname {arccsc}(x)+\operatorname {arcosh} |x|+C}

Fungsi hiperbolik

∫ sinh ⁡ x d x = cosh ⁡ x + C {\displaystyle \int \sinh x\,dx=\cosh x+C}

∫ cosh ⁡ x d x = sinh ⁡ x + C {\displaystyle \int \cosh x\,dx=\sinh x+C}

∫ tanh ⁡ x d x = ln ⁡ | cosh ⁡ x | + C {\displaystyle \int \tanh x\,dx=\ln |\cosh x|+C}

∫ coth ⁡ x d x = ln ⁡ | sinh ⁡ x | + C {\displaystyle \int \coth x\,dx=\ln |\sinh x|+C}

∫ sech x d x = arctan ⁡ ( sinh ⁡ x ) + C {\displaystyle \int {\mbox{sech}}\,x\,dx=\arctan(\sinh x)+C}

∫ csch x d x = ln ⁡ | tanh ⁡ x 2 | + C {\displaystyle \int {\mbox{csch}}\,x\,dx=\ln \left|\tanh {x \over 2}\right|+C}

Fungsi hiperbolik terbalik

∫ arsinh ⁡ x d x = x arsinh ⁡ x − x 2 + 1 + C {\displaystyle \int \operatorname {arsinh} x\,dx=x\operatorname {arsinh} x-{\sqrt {x^{2}+1}}+C}

∫ arcosh ⁡ x d x = x arcosh ⁡ x − x 2 − 1 + C {\displaystyle \int \operatorname {arcosh} x\,dx=x\operatorname {arcosh} x-{\sqrt {x^{2}-1}}+C}

∫ artanh ⁡ x d x = x artanh ⁡ x + 1 2 log ⁡ ( 1 − x 2 ) + C {\displaystyle \int \operatorname {artanh} x\,dx=x\operatorname {artanh} x+{\frac {1}{2}}\log {(1-x^{2})}+C}

∫ arcoth d x = x arcoth ⁡ x + 1 2 log ⁡ ( x 2 − 1 ) + C {\displaystyle \int \operatorname {arcoth} \,dx=x\operatorname {arcoth} x+{\frac {1}{2}}\log {(x^{2}-1)}+C}

∫ arsech x d x = x arsech ⁡ x − arctan ⁡ ( x x − 1 1 − x 1 + x ) + C {\displaystyle \int \operatorname {arsech} \,x\,dx=x\operatorname {arsech} x-\arctan {\left({\frac {x}{x-1}}{\sqrt {\frac {1-x}{1+x}}}\right)}+C}

∫ arcsch x d x = x arcsch ⁡ x + log ⁡ [ x ( 1 + 1 x 2 + 1 ) ] + C {\displaystyle \int \operatorname {arcsch} \,x\,dx=x\operatorname {arcsch} x+\log {\left[x\left({\sqrt {1+{\frac {1}{x^{2}}}}}+1\right)\right]}+C}

Integral lain, yaitu "Sophomore's dream", diyakini berasal dari Johann Bernoulli. Integral tersebut di antaranya

∫ 0 1 x − x d x = ∑ n = 1 ∞ n − n ( = 1 , 29128599706266 … ) ∫ 0 1 x x d x = − ∑ n = 1 ∞ ( − n ) − n ( = 0 , 78343051071213 … ) {\displaystyle {\begin{aligned}\int _{0}^{1}x^{-x}\,dx&=\sum _{n=1}^{\infty }n^{-n}&&(=1,29128599706266\dots )\\\int _{0}^{1}x^{x}\,dx&=-\sum _{n=1}^{\infty }(-n)^{-n}&&(=0,78343051071213\dots )\end{aligned}}}

Lihat pula

• Integral
• Kalkulus
• Fungsi gamma tidak komplit
• Jumlah tak terbatas
• Daftar limit
• Daftar deret matematikal
• Integrasi simbolik

Referensi

Pustaka

• M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.
• I.S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Errata. (Several previous editions as well.)
• A.P. Prudnikov (А.П. Прудников), Yu.A. Brychkov (Ю.А. Брычков), O.I. Marichev (О.И. Маричев). Integrals and Series. First edition (Russian), volume 1–5, Nauka, 1981−1986. First edition (English, translated from the Russian by N.M. Queen), volume 1–5, Gordon & Breach Science Publishers/CRC Press, 1988–1992, ISBN 2-88124-097-6. Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
• Yu.A. Brychkov (Ю.А. Брычков), Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008, ISBN 1-58488-956-X.
• Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002. ISBN 1-58488-291-3. (Many earlier editions as well.)

Sejarah

• Meyer Hirsch, Integraltafeln, oder, Sammlung von Integralformeln (Duncker und Humblot, Berlin, 1810)
• Meyer Hirsch, Integral Tables, Or, A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of Integraltafeln]
• David Bierens de Haan, Nouvelles Tables d'Intégrales définies (Engels, Leiden, 1862)
• Benjamin O. Pierce A short table of integrals – revised edition (Ginn & co., Boston, 1899)

Pranala luar

Tabel integral

• Paul's Online Math Notes
• A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More Exotic Functions): Indefinite Integrals Definite Integrals
• Math Major: A Table of Integrals Pranala usang diarsipkan 2012-10-30 di archive.today
• O'Brien, Francis J. Jr. "500 Integrals". Derived integrals of exponential and logarithmic functions
• Rule-based Mathematics Precisely defined indefinite integration rules covering a wide class of integrands
• Mathar, Richard J. (2012). "Yet another table of integrals". arXiv:1207.5845.{{cite arXiv}}: Pemeliharaan CS1: kekurangan kelas (link)

Derivasi

• V. H. Moll, The Integrals in Gradshteyn and Ryzhik

Layanan daring

• Integration examples for Wolfram Alpha

Program open source

• wxmaxima gui for Symbolic and numeric resolution of many mathematical problems Diarsipkan 2011-03-20 di Wayback Machine